Simulations of estimating causal effects using undersmoothed HAL
Seraphina Shi
2022-09-06
Simulation #1
Data structure: \(O = (W, A, Z, Y)\)
- U - exogenous variables
- W - baseline covariate that is a measure of body condition
- A - treatment level based on W, continuous between 0 and 5
- Z - intermediate curve based on W and A
- Y - outcome, indicator of an event ?
Underlying data generating process, \(P_{U,X}\)
- Exogenous variables:
- \(U_A \sim Normal(\mu=0, \sigma^2 =
1^2)\)
- \(U_A \sim Normal(\mu=0, \sigma^2 =
2^2)\)
- \(U_Z \sim Uniform(min = 0, max =
1)\)
- \(U_Y \sim Uniform(min = 0, max = 1)\)
- \(U_A \sim Normal(\mu=0, \sigma^2 =
1^2)\)
- Structural equations F and endogenous variables:
- \(W = U_W\)
- \(A = bound(2 - 0.5W + U_A, min=0,
max=5)\)
- \(Z = \mathbf{I}[U_Z < expit(2-W-A)]\)
- \(Y = \mathbf{I}[U_Y < expit(W + 5*A + Z - 0.5 * W * A - 8)]\)
- \(W = U_W\)
## W A Z Y
## Min. :-3.829933 Min. :0.0000 Min. :0.0000 Min. :0.0000
## 1st Qu.:-0.662628 1st Qu.:0.6306 1st Qu.:0.0000 1st Qu.:0.0000
## Median :-0.004864 Median :2.0093 Median :0.0000 Median :1.0000
## Mean :-0.002210 Mean :2.1248 Mean :0.4932 Mean :0.5977
## 3rd Qu.: 0.660425 3rd Qu.:3.4035 3rd Qu.:1.0000 3rd Qu.:1.0000
## Max. : 4.311039 Max. :5.0000 Max. :1.0000 Max. :1.0000## Summary of A given W < -1:## Min. 1st Qu. Median Mean 3rd Qu. Max.
## 0.000 1.407 2.798 2.689 4.029 5.000## Summary of A given -1 < W <= 0:## Min. 1st Qu. Median Mean 3rd Qu. Max.
## 0.0000 0.9264 2.2402 2.3016 3.5982 5.0000## Summary of A given 0 < W <= 1:## Min. 1st Qu. Median Mean 3rd Qu. Max.
## 0.0000 0.4099 1.7462 1.9385 3.1730 5.0000## Summary of A given 1 < W:## Min. 1st Qu. Median Mean 3rd Qu. Max.
## 0.000 0.000 1.255 1.574 2.746 5.000
##
## Call:
## glm(formula = Y ~ W + A + W * A + Z, family = binomial, data = obs)
##
## Deviance Residuals:
## Min 1Q Median 3Q Max
## -3.0483 -0.0492 0.0004 0.0467 3.7830
##
## Coefficients:
## Estimate Std. Error z value Pr(>|z|)
## (Intercept) -8.6327 0.2789 -30.952 <2e-16 ***
## W 0.6968 0.2233 3.121 0.0018 **
## A 5.3060 0.1566 33.877 <2e-16 ***
## Z 1.1786 0.1234 9.551 <2e-16 ***
## W:A -0.2709 0.1444 -1.876 0.0606 .
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## (Dispersion parameter for binomial family taken to be 1)
##
## Null deviance: 13478.7 on 9999 degrees of freedom
## Residual deviance: 2407.3 on 9995 degrees of freedom
## AIC: 2417.3
##
## Number of Fisher Scoring iterations: 9## [1] " MSE: 365.4844, AUC: 0.9908"
n = 50
CV HAL
results
## CV selected lambda (from one sample): 0.00990300430080305
1000 repetition
## The average of CV selected lambdas (from 1000 sample): 0.0177983801957674 The average of CV selected lambdas (from 1000 sample): 0.0177862720162932 The average of CV selected lambdas (from 1000 sample): 0.0178028065364971 The average of CV selected lambdas (from 1000 sample): 0.0177907302552206## z=1:
## z=0:
Globally Undersmoothed HAL
results
## Undersmoothed lambda: 0.00037559027270435
## which is 0.0379269019073225 * lambda_CV
1000 repetition
## The average of unsersmoothed lambda (from 1000 sample): 0.000581076546605094
## which is 0.0329348525092873 * the average of 1000 lambda_CV The average of unsersmoothed lambda (from 1000 sample): 0.000580903922299673
## which is 0.0329303125559549 * the average of 1000 lambda_CV The average of unsersmoothed lambda (from 1000 sample): 0.000580778134680317
## which is 0.0329164209081906 * the average of 1000 lambda_CV The average of unsersmoothed lambda (from 1000 sample): 0.000580426573286071
## which is 0.0329164209081906 * the average of 1000 lambda_CV## z=1:
## z=0:
Oevr a grid of lambda scalers
## TableGrob (7 x 4) "arrange": 7 grobs
## z cells name grob
## 1 1 (2-3,2-3) arrange gtable[layout]
## 2 2 (4-5,1-2) arrange gtable[layout]
## 3 3 (4-5,3-4) arrange gtable[layout]
## 4 4 (6-7,1-2) arrange gtable[layout]
## 5 5 (6-7,3-4) arrange gtable[layout]
## 6 6 (3-3,4-4) arrange gtable[guide-box]
## 7 7 (1-1,1-4) arrange text[GRID.text.1916]
## TableGrob (7 x 4) "arrange": 7 grobs
## z cells name grob
## 1 1 (2-3,2-3) arrange gtable[layout]
## 2 2 (4-5,1-2) arrange gtable[layout]
## 3 3 (4-5,3-4) arrange gtable[layout]
## 4 4 (6-7,1-2) arrange gtable[layout]
## 5 5 (6-7,3-4) arrange gtable[layout]
## 6 6 (3-3,4-4) arrange gtable[guide-box]
## 7 7 (1-1,1-4) arrange text[GRID.text.2192]
## TableGrob (7 x 4) "arrange": 7 grobs
## z cells name grob
## 1 1 (2-3,2-3) arrange gtable[layout]
## 2 2 (4-5,1-2) arrange gtable[layout]
## 3 3 (4-5,3-4) arrange gtable[layout]
## 4 4 (6-7,1-2) arrange gtable[layout]
## 5 5 (6-7,3-4) arrange gtable[layout]
## 6 6 (3-3,4-4) arrange gtable[guide-box]
## 7 7 (1-1,1-4) arrange text[GRID.text.2468]